modeling wheel-induced rutting in soils: rolling · 2013. 1. 22. · theoretical models for...

Available online at www.sciencedirect.comJournal

www.elsevier.com/locate/jterra

Journal of Terramechanics 46 (2009) 35–47

ofTerramechanics

Modeling wheel-induced rutting in soils: Rolling

J.P. Hambleton, A. Drescher *

Department of Civil Engineering, University of Minnesota, 500 Pillsbury Drive SE, Minneapolis, MN 55455, USA

Received 17 June 2008; received in revised form 19 February 2009; accepted 25 February 2009Available online 3 April 2009

Abstract

Theoretical models for predicting penetration of non-driving (towed) rigid cylindrical wheels rolling on frictional/cohesive soils arepresented. The models allow for investigating the influence of soil parameters and wheel geometry on the relationship between theinclined rolling force and wheel sinkage in the presence of permanently formed ruts. The rolling process is simulated numerically in threedimensions using the finite element code ABAQUS. The numerical simulations reveal that the advanced three-dimensional process ofrutting can be regarded as steady, and an approximate analytic model for predicting sinkage under steady-state conditions, whichaccounts for three-dimensional effects, is also developed. The differences between wheel rolling and wheel indentation (considered ina separate paper) are discussed. Numerical and analytic results are compared with test results available in the literature and obtainedfrom preliminary small-scale experiments, and general agreement is demonstrated.� 2009 ISTVS. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Models for predicting soil rutting induced by a rollingwheel can be used to determine the impact of off-road vehi-cles (ORVs) in sensitive natural areas (cf. [1]), assess mobil-ity of ORVs in adverse terrain, and facilitate methods forevaluating in situ soil properties premised on relating rutdepth to material strength parameters [2]. Formation ofpermanent ruts by the rolling wheels of ORVs presents aparticularly complex and challenging problem when ana-lyzed within the framework of mechanics. The primary rea-son is that rutting is a result of a process rather than aparticular state of loading, with the latter often being suf-ficient in the study of geomechanics problems. When rut-ting occurs, the soil undergoes a loading–unloadingsequence resulting from wheel contact that ultimatelyinduces extremely large and inherently three-dimensionaldeformation.

In a separate paper [3], the initial phase of rutting wasmodeled as an indentation process in which the wheel dis-

0022-4898/$36.00 � 2009 ISTVS. Published by Elsevier Ltd. All rights reserve

doi:10.1016/j.jterra.2009.02.003

* Corresponding author. Tel.: +1 612 625 2374; fax: +1 612 626 7750.E-mail addresses: [email protected] (J.P. Hambleton), dresc001@

umn.edu (A. Drescher).

places normally into the soil without rotation or horizontal(i.e., longitudinal) translation of the wheel. This paper isdedicated to modeling the formation of a rut in the processof a wheel rolling on soil and, in particular, the advancedphase when the process can be regarded as steady. In thesteady configuration, stresses and the velocity field whenmeasured with respect to the wheel do not change withtime. Both papers have their origin in the exploratory workpresented in [2,4] and complement numerous monographsand papers in the literature on the subject (e.g., [5–13]).All phases of rutting are considered here as quasi-static,which is acceptable for wheels traveling with slow or mod-erate velocities. The modeling is further limited to non-driving wheels, which do not transmit a torque. Non-driv-ing wheels include the front wheels of many vehicles andwheels of towed equipment. This assumption simplifiesthe analysis, whereas slip and erosion induced by drivingwheels complicates the problem significantly.

The most essential characteristic of wheel-induced rut-ting is the three-dimensionality of the deformation field,which involves only one plane of symmetry parallel tothe midplane of the rolling wheel. The deformation fielddepends on the type of soil and its previous loading history,which affect possible soil distortional and volumetric

d.

mailto:[email protected]

mailto:dresc001@ umn.edu

mailto:dresc001@ umn.edu

36 J.P. Hambleton, A. Drescher / Journal of Terramechanics 46 (2009) 35–47

changes. Material is typically pushed in front of and to thesides of the wheel, which results in the formation of twoparallel berms along an extending rut (Fig. 1a). Rigoroussolutions for this complex three-dimensional problem can-not be obtained analytically, leaving numerical simulationas the only viable means for precise modeling. Valuableexamples of three-dimensional numerical modeling aimedat simulating wheel penetration in soils or snow [14,15]and modeling the interaction of various tools with soils[16–18] can be found in the literature. Attempts also havebeen undertaken to consider rolling as a two-dimensionalprocess [19–23].

The present paper concentrates on exploring the ade-quacy of modeling rutting with a simple elastic–plastic con-stitutive law, consisting of a linear elastic part (Young’smodulus E; Poisson’s ratio m) and perfectly-plastic partdescribed by the Mohr–Coulomb yield condition (frictionangle u; cohesion c) with an associated or non-associatedflow rule (dilation angle w 6 u). Elastic parameters playa secondary role in the analysis and were fixed atE/cd = 1000 and m = 0.3, where wheel diameter d and soilunit weight c are used for normalization. This elastic–plas-tic law was employed in preliminary analyses described in[2,4] and the detailed analysis of indentation [3] in whichpredicted soil response displayed encouraging agreementwith experimental data. A tacit assumption reflected inthe soil constitutive models used in this paper is that thesoil does not compact. Ample literature is available toshow the relevance of compaction for loose materials(e.g., [24–28]), although compaction is minimal or alto-gether absent for a large class of materials including satu-rated clays and dense sands.

The main objective of the theoretical modeling pre-sented in this paper was to arrive at a relationship betweenwheel load and penetration that incorporates wheel geom-etry and basic soil properties, considering the simplest caseof a rigid, right-cylindrical wheel. Wheel sinkage, denoteds, results from both inelastic and elastic soil deformationand in general is not the same as the rut depth r, withs P r (Fig. 1b). The difference between s and r arises duepartly to elastic rebound of the soil and partly to upwardplastic flow in a zone directly to the rear of the wheel, ascan be seen in images of deformation induced by a rolling

Fig. 1. Deformation induced by non-driving wheel: (a) photograph of wheelwheel forces.

cylinder [7]. However, it is shown in this paper throughcomparison of s and r computed from numerical simula-tion that sinkage and rut depth can be regarded as practi-cally interchangeable (s � r) for a non-driving wheel andthe soil model considered. Since s is easily determined fora rigid wheel, being equivalent to vertical wheel displace-ment, it is the variable predominantly used in this paper.

As in [2–4], numerical simulations presented in thispaper were performed using the explicit (dynamic) versionof the computational platform ABAQUS. The numericalsimulations are supplemented by an approximate analyticapproach originally presented in [2,4] and expanded in[3]. The results of computations are compared with exper-imental data available in the literature and exploratorysmall-scale experiments. Finally, the similarities and differ-ences between wheel rolling and wheel indentation areassessed.

2. Numerical simulations

The very large deformations induced by a rolling wheelmake the ABAQUS/Standard (Lagrangian) version of theABAQUS code inadequate for numerical simulations dueto unacceptable distortions of the initial finite elementmesh. An alternative used in this paper is the ABAQUS/Explicit version [29], which provides an arbitrary Lagrang-ian/Eulerian (ALE) analysis option allowing for continu-ous remeshing. This version also was used in [3] foranalyzing wheel indentation, thus providing data for com-parison that is unaffected by algorithmic differences. Use ofABAQUS/Explicit required approximating the Mohr–Coulomb yield condition by the extended Drucker–Prageryield condition [29], which may be considered a smoothcounterpart of the Mohr–Coulomb criterion and has a cor-responding associated or non-associated flow potential.The two conditions match exactly in stress states corre-sponding to triaxial compression and are in close agree-ment in triaxial extension.

In contrast to wheel indentation, the mesh considered inrolling consisted of two regions with different materialproperties. This was motivated by the fact that the forma-tion of a rut begins when a wheel passes from a relativelystrong soil onto a weak bed. To simulate this process, the

on medium density sand; (b) schematic depicting sinkage, rut depth, and

J.P. Hambleton, A. Drescher / Journal of Terramechanics 46 (2009) 35–47 37

soil volume was partitioned into regions of strong andweak material (Fig. 2), with material properties for thestrong region consistently chosen such that little wheel pen-etration occurred. Both regions were discretized using 8-node linear hexahedral elements with reduced integrationand hourglass control. The number of elements wasroughly 65,000 in total. Out-of-plane displacements werenot allowed on all boundaries except the free surface atwhich the wheel and soil interact, thus providing the sym-metry condition on the boundary representing the mid-plane of wheel (the face of the soil domain visible inFig. 2 and parallel to the y–z plane) and giving remainingboundaries limited freedom to displace. Boundaries ofthe soil volume were placed far enough from the locationof the wheel to have virtually no effect on the deformationfield or rolling horizontal force.

The right-cylindrical wheel of diameter d and width b

was modeled as an analytical rigid surface (i.e., not discret-ized) and governed by a single reference node located at thewheel center. The edge of the wheel was filleted, with a filletradius of b/20. Introducing a fillet was necessary to avoidnumerical problems in the algorithm reproducing changingcontact between the rotating wheel and the soil. The soil–wheel interface was assumed to be frictional, with coeffi-cient of friction l = 0.5.

The simulations began with application of a uniformbody force over the entire soil region corresponding tothe soil unit weight c. Next, a concentrated vertical(z-direction in Fig. 2) force QV representative of weighton the wheel was applied gradually at the wheel referencenode while the wheel was positioned on the strong soilregion, and as a result the wheel settled slightly into thesoil. After application of QV, the wheel reference nodewas given a horizontal (y-direction) linear velocity, whichramped up slowly to a specified value and became constantwhen the center of the wheel was directly above the bound-ary between the strong and weak soil regions. When mov-ing horizontally, the wheel was free to rotate about itscenter and move vertically while the vertical force and hor-

Fig. 2. Reference configuration for numerical simulation of rolling wheel(wheel travels in positive y-direction).

izontal velocity were kept constant. Wheel rotation was aresult of frictional interaction at the soil–wheel contact sur-face. The initial vertical loading rate and horizontal veloc-ity were chosen small enough to guarantee negligibleinertial forces, and adaptive meshing was used to avoidexcessive element distortion.

During rolling, evolution of the horizontal componentof the force on the wheel, denoted QH, and vertical wheeldisplacement, equivalent to sinkage s for the rigid wheel,were the particular variables of interest. Field variables inthe soil region were also readily determined from the sim-ulations, with the displacement field being of interest forcomparison with experimental data. Applied vertical forceswere such that computed values of steady-state sinkagewere moderate (i.e., s/d < 0.1) and representative of thosecommonly encountered in application.

Pervasive numerical instabilities were encountered whenattempting to simulate a wheel rolling on purely frictionalsoil, much like in the case of wheel indentation [3] but to agreater extent. For indentation, stability was maintained byintroducing small cohesion. A similar approach wasemployed in the simulations of rolling, with the markeddifference that the cohesion necessary for stability in therolling case was quite large. The instabilities appear toresult from the code’s inability to perform calculationsfor stress states in the neighborhood of the yield condi-tion’s vertex, which corresponds to zero isotropic stressfor cohesionless material. Unlike indentation in which iso-tropic stresses are primarily compressive, the deformedconfiguration for rolling is such that stress states are consis-tently near or at the yield condition’s vertex, particularly atthe soil surface directly to the rear of the wheel. Likewise,granular materials undergo avalanching as material col-lapses into the rut left by a wheel or is pushed in front ofthe wheel, and this phenomenon cannot be realisticallyreproduced by a continuum model.

3. Results of numerical simulations

Fig. 3 is an example of the deformed mesh at the end ofa simulation with purely cohesive (clay) soil (u = w = 0;plastic incompressibility). Normalized cohesion in the sim-ulation was c/cd = 1.25, and a normalized vertical forceQV/cbd2 = 1.9 was applied to a wheel with aspect ratiob/d = 0.3. Variation of s and QH with horizontal wheel dis-placement uy is shown in Fig. 4. As seen in this figure, sink-age and horizontal force become nearly constant after aninitial transient phase of rolling (roughly 0 6 uy/d 6 1.5).This advanced phase, in which there is material upheavedin front of the wheel and a rut surrounded by two bermsto the rear of the wheel (Fig. 3), can thus be regarded assteady-state operation of the wheel.

The existence of a steady-state was revealed for otherwheel aspect ratios, although steady-state for very widewheels (e.g., b/d > 1.5) and moderate to large wheel forces(QV/cbd2 > 1) was not readily achieved within the rollingdistance uy/d 6 2.25 allowed by the reference configuration

Fig. 3. Deformed mesh at end of simulation with cohesive soil (u = w = 0, c/cd = 1.25, b/d = 0.3, QV/cbd2 = 1.9): (a) side view (direction of wheel travel isfrom left to right); (b) view from front of the wheel.

Fig. 4. Sinkage and horizontal wheel force vs horizontal wheel displace-ment for simulation with cohesive soil (u = w = 0, c/cd = 1.25, b/d = 0.3,QV/cbd2 = 1.9).

Fig. 5. Vertical force vs steady-state sinkage with varying wheel aspectratios and cohesive soil (u = w = 0, c/cd = 1.25).


considered. The differing behavior for wide wheels isexplained by the tendency for material to accumulate infront of the wheel rather than move to the sides. This ‘‘bull-dozing” effect for wide wheels has been noted in severalrolling resistance studies involving various materials[5,30,31]. Gee-Clough [31] showed a photograph of mate-rial accumulating to a height of roughly d/2 for a kinemat-ically-controlled wheel (s/d = 0.25) with b/d = 0.54operating on sand. For the sake of comparison, the maxi-mum aspect ratio considered in this paper is b/d = 1.2,although such a roller geometry should be considered a cyl-inder, or drum, rather than a wheel. Typical aspect ratiosfor wheels of SUVs, ATVs, special-purpose ORVs, andoff-road motorcycles are 0.1 6 b/d 6 0.5.

The results of a single three-dimensional simulation, asin Figs. 3 and 4, provide steady-state values of s and QH

for one particular value of QV. Several simulations withdifferent vertical force were thus required to obtain the glo-bal force–sinkage relationship for specified material prop-erties and wheel geometry. The resulting steady-stateforce–sinkage curves for cohesive soil and some wheelaspect ratios are shown in Fig. 5. As in remaining figures,a smooth approximation to the discrete data is includedto highlight trends. Clearly, the steady-state force–sinkagerelationship is nonlinear, with the slope of the curves

decreasing with increasing sinkage, and vertical force perunit width increases with increasing aspect ratio. Fig. 6depicts the dependence of the force inclination angle b ons under steady-state conditions, where b is defined as(Fig. 1b)

b ¼ tan�1 QH

QV

� �ð1Þ

It may be noted that plotting b as a function of s is basi-cally equivalent to showing QH as a function s, with theformer being a more convenient representation for the pur-poses of this paper. As a result of the increased accumula-tion in front of a relatively wide wheel, angle b increasessharply with an increase in b/d.

Steady-state force–sinkage curves resulting from simula-tions for a frictional soil with sufficient cohesion for com-putational stability (u = 30�, w = 0, c/cd = 0.25) areshown in Fig. 7. The curves exhibit similar trends as seenfor cohesive material except that force per unit width isnot strictly increasing with increasing aspect ratio, beingfor the most part larger with b/d = 0.1 than with b/d = 0.3. The relationship between steady-state b and s forvarying b/d was found to be virtually identical to thatdetermined for the cohesive material with u = w = 0 andc/cd = 1.25 (Fig. 6). Increasing dilation angle had a similareffect on QV as in indentation [3], with both QV and b

Fig. 6. Force inclination angle vs sinkage in steady-state with varyingwheel aspect ratios and cohesive soil (u = w = 0, c/cd = 1.25).

Fig. 7. Vertical force vs steady-state sinkage with varying wheel aspectratios and frictional/cohesive soil (u = 30�, w = 0, c/cd = 0.25).

Fig. 8. Steady-state sinkage and rut depth from numerical simulationswith varying soil properties (QV/cbd2 = 7, b/d = 0.3).


increasing substantially with increasing w. Also like inden-tation, the coefficient of interface friction l P 0 had rela-tively little effect on the force–sinkage curves.

In addition to sinkage, rut depth can be readily deter-mined from numerical simulations. Fig. 8 compares sink-age and rut depth for a heavily-loaded wheel (QV/cbd2 =7, b/d = 0.3) in steady rolling on various soils. It is seen thatrut depth and sinkage are virtually the same when wheelpenetration is significant, with elastic deformation causingsome difference at small sinkage (i.e., for strong materials).

In the next section, an analytic approach for predictingsteady-state wheel sinkage is developed. This method notonly provides closed-form formulae but also allows for pre-dicting sinkage on purely frictional soils.

4. Approximate analytic approach

The approximate analytic approach bears some similar-ity to the analysis of tillage forces considered in [32] and

focuses on determining the force on a non-driving wheelin an advanced, steady state of rolling. The approach isan extension of the methodology discussed for indentation[3] and derives from preliminary work presented in [2,4].The method hinges on two fundamental assumptions.The first is that the actual geometry of the soil–wheel con-tact interface can be considered equivalent to a flat rectan-gular surface with area and inclination determineduniquely by the sinkage. Accordingly, the approach applieswhen sinkage is small relative to the wheel diameter. Thesecond assumption is that the yielding state in the soil issuch that average stress over the flat rectangular surfacemay be taken as the average ultimate stress (bearing capac-ity) for an equivalent shallow foundation. Bearing capacityis denoted qu and calculated based on the Terzaghi–Meyer-hof formula [33,34]

qu ¼ cN cF csF cd F ci þ qNqF qsF qdF qi þ1

2cBN cF csF cdF ci ð2Þ

where q = cD is surcharge acting at footing depth D and Bis the footing width. Formulas for the bearing capacity fac-tors Nc, . . .,Nc, shape factors Fcs, . . .,Fcs, depth factorsFcd, . . .,Fcd, and inclination factors Fci, . . .,Fci appearingin Eq. (2) are given in Appendix A.

After computing qu according to Eq. (2), wheel forcecorresponding to given sinkage is determined as the prod-uct of qu and the assumed soil–wheel contact area. The for-mulation is fully three-dimensional in that the shapefactors in Eq. (2) account for changes in average stress aris-ing due to the aspect ratio of the footing B/L, where L P B

is the footing length. Unlike the numerical simulations,dilation angle w nowhere directly enters the calculation,although the bearing capacity formula was derived semi-analytically for Mohr–Coulomb material with associatedplastic flow (w = u).

The approximate analytic method is predicated onknowing a priori the force inclination angle b. This angleis related to the distribution of contact stresses over the

Fig. 10. Comparison of force inclination angle from Onafeko and Reece[36], numerical simulations (steady-state values), and Eq. (4).


soil–wheel interface and therefore can be assessed fromexperimental stress measurements [35–37]. An example ofthe contact stresses measured by Onafeko and Reece [36]for a non-driving wheel is shown in Fig. 9. Based on thesemeasurements, it is now assumed that the distribution ofshear stress is antisymmetric about the ray originating fromthe wheel center and bisecting the arc corresponding to thesoil–wheel contact interface. Likewise, it is assumed thatthe normal stress distribution is symmetric. From theseassumptions, it follows that b is simply half of the contactangle a shown in Fig. 1b. Since a is geometrically related tos and d, angle b may be expressed as

b ¼ a2¼ 1

2cos�1 1� 2

sd

� �ð3Þ

Keeping only the first term of a power series expansion of babout s/d = 0 gives

b ¼ffiffiffisd

rð4Þ

Eq. (4) is quite similar to expressions derived empirically orsemi-empirically in previous works [5,31,38], in which theratio QH/QV was expressed as simple or relatively compli-cated functions of

ffiffiffiffiffiffiffis=d

p.

Fig. 10 compares predicted values of b using Eq. (4) withresults from numerical simulations. The equation qualita-tively matches the numerical results very well, with b beingsomewhat underestimated with relatively large s. Thenumerical results predict that b depends to some extenton b/d, but agreement is reasonable for wheels of typicalwidth. Values from numerical simulations for cohesive soil(u = w = 0, c/cd = 1.25) and frictional/cohesive soil(u = 30�, w = 0, c/cd = 0.25) are virtually indistinguish-able, supporting the material-independence of Eq. (4). Alsoshown in the Fig. 10 are experimentally-determined pointscalculated from values of QV, QH, and s reported in [36],which agree very well with predictions from Eq. (4).

Two versions of the analytic approach, which handle theeffects of inclined loading in different ways, are nowconsidered.

4.1. Inclined force method

In the first version of the analytic approach, the wheel iseffectively replaced by a horizontal footing, and the totalforce Q is therefore inclined relative to the footing

Fig. 9. Contact stresses measured by Onafeko and Reece [36] for towedrigid wheel with d = 1.2 m and b = 0.3 m on loose sand.

(Fig. 11c). This formulation, referred to as the ‘‘inclinedforce method”, therefore accounts for the effect of inclina-tion directly through the inclination factors Fci, . . .,Fci

appearing in Eq. (2). In keeping with considering the foot-ing to be horizontal, the footing contact distance h is takenas shown in Fig. 11a and is geometrically related to sinkageand wheel diameter as

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

ð5ÞThe footing width B and length L in the bearing capacityformula are then

B ¼ h

L ¼ b

�for h < b;

B ¼ b

L ¼ h

�for h � b ð6Þ

and the total force Q together with its vertical componentQV are

Q ¼ qubh;QV ¼ Q cos b ¼ qubh cos b ð7Þ

Fig. 11. Rolling wheel in steady-state as analogous shallow footing: (a)side view of wheel; (b) plan view of wheel (direction of wheel travel is fromleft to right); (c) equivalent footing with inclined force; (d) equivalentinclined footing.

Fig. 12. Examples of vertical force vs sinkage using analytic (inclinedfooting) method with b/d = 0.2.

Fig. 13. Force–sinkage predictions using analytic method and numericalsimulations with cohesive soil (u = w = 0, c/cd = 1.25).


The variable D appearing in (2) is viewed as accounting formaterial pushed in front and along the sides of the wheel,and it enters through the surcharge q = cD and the depthfactors Fcd, . . .,Fcd. Depth D is estimated by consideringmass balance of soil displaced by the wheel [2], resultingin the approximation

D ¼ 1

6s ð8Þ

Final algebraic expressions relating QV to s, b, d, u, c, and care determined by substituting results (2), (4), (5), (6), and(8) into Eq. (7). The resulting equations are in Appendix A.

4.2. Inclined footing method

A second version of the analytic method is developed byreplacing the wheel with an equivalent footing that isinclined at angle b to the horizontal (Fig. 11d). This versionis referred to as the ‘‘inclined footing method”. With thefooting inclined at angle b, the total force Q is necessarilyperpendicular to the footing, and the configuration isnearly identical to the case of a vertically loaded footingon a slope. Bearing capacity for a footing on slopingground can be calculated with formula (2) using modifiedbearing capacity factors N �c , N �q, and N �c . The modified fac-tors proposed in [39,40], given in Appendix A, are used infurther developments. An error is introduced because themodified bearing capacity factors were derived for gravityacting perpendicular to the footing, although this error isminimal for small sinkage. Also, shape and depth factorsare taken to be the same as for a footing on horizontalground, despite the fact that they are somewhat differentfor sloping ground.

In addition to the modified bearing capacity factors, theinclined footing method differs in how the footing lengthand equivalent depth D are calculated. The variable h inthe inclined force method is replaced with h� (Fig. 11a),where

h� ¼ffiffiffiffiffidsp

ð9ÞDepth D is also replaced with D� to be consistent with theconcept of a footing on sloping ground:

D� ¼ D cos b ¼ 1

6s cos b ð10Þ

The final force–sinkage equations resulting from the in-clined footing method are in Appendix A.

5. Results of approximate analytic approach

Force–sinkage curves predicted using the analyticapproach are similar in character to the numerical results.Fig. 12 shows example curves using the inclined footingmethod for varying soil properties and b/d = 0.2. Sinkageat which B = L is clearly visible in the figure, as the firstderivative of QV with respect to s is discontinuous at thispoint. As with the curves from numerical simulation, verti-

cal force is a nonlinear function of sinkage, with the forceincreasing at a decreasing rate. For cohesive soil (u = 0),force increases roughly linearly with cohesion, and for fric-tional soil (c = 0), force increases almost exponentially withfriction angle.

Fig. 13 compares results from numerical simulation withthose from the analytic approach for cohesive soil. Theinclined footing method gives curves situated above thosefrom the inclined force method, and they locate closer tocurves obtained from numerical simulations. The analyticapproach qualitatively predicts the same dependence offorce on the wheel aspect ratio (i.e., three-dimensionaleffects) as the numerical simulations. The overall shape ofthe curves resulting analytically and numerically is similar,with better agreement at small sinkage than large sinkage.

Fig. 14 compares analytic and numerical results for fric-tional/cohesive soil. Trends are similar to the case of cohe-sive soil, and the inclined footing method again givespredictions closer to numerical simulations than the

Fig. 15. Steady-state incremental displacement field in granular materialapproximating response at wheel midplane (direction of wheel travel isfrom right to left; numerals on scale denote cm): (a) experimental resultsusing PIV; (b) results of numerical simulation (u = 40�, w = 20�,c/cd = 20).

Fig. 14. Force–sinkage predictions using analytic method and numericalsimulations with frictional/cohesive soil (u = 30�, w = 0, c/cd = 0.25).


inclined force method. Agreement between analytic andnumerical predictions for b/d = 0.3 is quantitatively quitegood, whereas the discrepancy is considerable for b/d = 0.1. Furthermore, the analytic approach and numericalsimulations contradict one another in terms of how force isaffected by b/d at relatively large sinkage, with numericalsimulations indicating that force per unit width should behigher for b/d = 0.1 than b/d = 0.3 and the analyticapproach predicting the opposite. The analytic approachdoes, however, predict the same trend as the numerical sim-ulations in the interval 0 6 s/d 6 0.025. Effects of b/d in theanalytic method are linked to the shape factors present inthe bearing capacity formula (Eq. (2)), and differences withnumerical results point to possible improvements to thesefactors (e.g., [41]).

6. Experiments

Simple constitutive models for the soil were used in thenumerical and approximate analytic approaches presentedin previous sections. To assess the capability of bothapproaches in predicting wheel sinkage in real soils, limitedexperimental data were generated using two setups similarto those used for indentation [3]. The first setup, consistingof a 780 � 440 � 80 mm container with a Plexiglas frontwall, provided information on the incremental displace-ment field in a granular material subject to wheel rolling.The material consisted of crushed walnut shells, character-ized in [3,42], over which an acrylic wheel of diameterd = 100 mm and width b = 19 mm was rolled next to thetransparent wall to simulate one-half of a three-dimen-sional rolling wheel with b/d = 0.38. The wheel wasattached to a horizontal shaft and rolled at prescribed sink-age (kinematic control), meaning that the wheel’s axis ofrotation was kept at the same position relative to the undis-turbed material surface as the wheel moved. The wheel wasfree to rotate (i.e., non-driving or towed wheel), with freerotation facilitated by leaving a very small gap between

the wheel and Plexiglas wall. Even though the frictionbetween the material and wall affected local displacements,it may be argued that the measurements represent quanti-ties at the midplane of a fully three-dimensional configura-tion reasonably well.

Fig. 15a is an example of the displacement incrementfield obtained using digital photography processed withparticle image velocimetry (PIV) software [43]. Prescribedsinkage was s/d = 0.04, with analysis of the deformationat several wheel locations indicating a nearly steady field.Accumulation in region BCDKB is clearly visible, andbecause the wheel is transparent, the height of materialflowing to the side of the wheel and forming a berm canbe seen. Particles move forward within zone ABCDEIJAin front of the wheel, and particles move rearward in regionEFGHIE below and to the rear of the wheel. RegionsABJA and EGHE are characterized by upward motion,and particles move downward in zone DEID. RegionsDIJD and EHIE are transition regions with both down-ward and upward flow. Overall, the mechanism looks likea skewed version of the classical Prandtl failure mechanismfor a shallow footing [44], with the key difference that nodistinct boundaries with a jump in displacement increments(shear bands [45] or shocks [46]) were observed. Vectors arenot shown in regions BCDKB and EFLE due to poorimage correlation in the PIV analysis, which appears tobe the result of avalanching and particle rotation in theseregions.


In view of failed numerical simulations of rolling onpurely frictional material, no direct comparison with exper-iments was possible. Instead, a simulation of rolling on africtional/cohesive material with u = 40�, w = 20�, and c/cd = 20 is shown in Fig. 15b. The friction and dilationangles are roughly the measured properties for the materialand gave numerical results agreeing favorably with theexperimentally-determined kinematics for indentation [3].It is evident that in spite of large cohesion the field of dis-placement increments resembles the experimental one, andthis is consistent with the generally-accepted observationfrom numerous geomechanics problems (e.g., retainingwalls and foundations) that cohesion does not affect thekinematics significantly in the region of induced shear.

Several features in the experiment are not, however,reproduced in the numerical simulation. Namely, numeri-cal results show no rearward flow zone and give that theheight of material pushed in front of the wheel is approxi-mately 1.5 times that measured experimentally. These dif-ferences appear to be caused by the cohesion required forcomputational stability. It is not surprising that a discrep-ancy arises in the zone directly below and to the rear of thewheel, as it is this region where tensile isotropic stressesdevelop in the numerical simulation despite the fact thatsuch stresses cannot exist in granular material. Similarly,avalanching of material in front of the wheel was observedin the experiments but does not occur in the simulation,with avalanching causing the height of material in frontof the wheel to decrease.

The second experimental setup was aimed at providingdata on the relationship between the vertical force on therolling wheel and sinkage in a true three-dimensional set-ting. These were conducted on saturated clay and dry sand(described in [3]) placed in a 100 � 250 � 300 mm con-tainer. The container was affixed to a linear bearing, anda wheel of diameter d = 115 mm and width b = 38 mm withcoarse sandpaper adhered to the contacting surface wasattached to a load cell connected to a vertical shaft abovethe center of the container. The load cell was oriented tomeasure the vertical component of force acting on thewheel, and it was verified that horizontal force on the wheel,which was not measured during the experiments, had an

Fig. 16. Comparison of experimental and predicted stead

insignificant affect on readings of vertical force. As withthe experiment involving PIV, the wheel was free to rotate,such that a non-driving wheel was replicated. The wheel waslowered to prescribed sinkage, and the container was thenmoved horizontally by means of a stepper motor. After ini-tial variation, the vertical force became nearly constant withfurther rolling of the wheel, thus indicating a state close tobeing steady. Several tests with different sinkage were con-ducted to construct the force–sinkage curves.

Experimental results are shown in Fig. 16 together withsuperimposed curves obtained from numerical simulationsand the analytic approach using u and c (as well as E innumerical simulation) as determined from uniaxial and tri-axial compression [3]. The theoretical predictions agreefavorably with the experimental results, particularly forthe sand. For the clay, both numerical and analytic predic-tions somewhat overestimate the wheel force at small sink-age, and the analytic method somewhat underestimates theforce at large sinkage. As results using the analyticapproach are based on the inclined footing method, itappears that the inclined footing method provides morerealistic predictions than the inclined force method.

Predictions using the analytic approach are also com-pared with force–sinkage data obtained by Willis et al.[47], who performed tests with rigid wheels on clay andsand. The test procedure consisted of rolling a wheel undergiven vertical load (static control) and measuring sinkage.Reported properties were u = 7� and c = 22 kPa for theclay. Properties for the sand were u = 35� andc = 14 kN/m3. It is not clear from the text how u and cwere determined by the authors, but the values are never-theless typical for clay with high water content and mediumdensity sand. Comparison is made with test results for awheel with d = 406 mm and b = 76 mm on clay and awheel with d = 406 mm and b = 114 mm on sand. Theexperimental data are shown in Fig. 17 and compared withtheoretical predictions using the analytic method. Verygood agreement is seen in the comparison for clay, and rea-sonable agreement is found for sand. Commensurate withthe uncertainty in measured soil properties, two theoreticalcurves are presented for the sand, which also show the sen-sitivity of the prediction with respect to u.

y-state force–sinkage curves: (a) clay; (b) dense sand.

Fig. 17. Comparison of force–sinkage predictions using analytic (inclined footing) method and data from Willis et al. [47]: (a) clay; (b) sand.


7. Comparison between rolling and indentation

Fig. 18a shows, based on numerical simulation withcohesive soil, the relationship between vertical force andsteady-state sinkage for rolling as compared with indenta-tion [3]. Qualitatively, the shape of the curves is very simi-lar; however, the magnitude of the force for a given sinkagein rolling is just over half of that for indentation. As nonumerical results were available for purely frictional mate-rial, only the approximate analytic results are compared(Fig. 18b). At given sinkage, vertical force for rolling isdrastically less than with indentation. These differencesbetween rolling and indentation for cohesive and frictionalsoils are further supported by the experimental resultsshown in Fig. 19. Obviously, there is a fundamental differ-

Fig. 18. Comparison of force–sinkage curves for steady rolling and indentatioc/cd = 1.25); (b) results from analytic method with frictional soil (u = 35�, c/c

Fig. 19. Comparison of force–sinkage curves for steady rolling and ind

ence in the deformation field in rolling and indentation,and therefore no direct comparison is made.

There are three essential differences between indentationand rolling of a non-driving wheel. The first basic differenceis that the soil–wheel contact area in the rolling process issmaller than with indentation at the same sinkage. Thisarea in rolling is approximately half of that in indentation,and this is clearly seen in the expressions used in the ana-lytic approach for the contact length h with rolling (Eqs.(5) and (9))

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

� h� ¼ffiffiffiffiffidsp

ðsmall s=dÞ ð11Þand with indentation [3]

h ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

ð12Þ

n [3]: (a) results from numerical simulation with cohesive soil (u = w = 0,d = 0, inclined footing method).

entation [3] from small-scale experiments: (a) clay; (b) dense sand.


If one regards vertical force as being fixed, the contact areareduction in rolling as compared to indentation requiresthat the wheel sinks deeper during rolling. The second keydifference is that the load is inclined in rolling, which alsocauses a rolling wheel to sink more than an indenting wheelwith the same vertical force. Lastly, rolling and indentationare distinguished by the tendency for material to accumu-late in front of a rolling wheel. The significance of accumu-lation can be shown by comparing the effects of the wheelaspect ratio for rolling and indentation with cohesive mate-rial. In rolling, vertical force per unit width was found to bestrictly increasing with increasing b/d (Fig. 5). In indenta-tion [3], vertical force per unit width for given sinkage wasfound to be relatively insensitive to b/d. The change independence on b/d between rolling and indentation is ex-plained by resistance from accumulated material in frontof the wheel during the rolling process.

8. Closing remarks

In numerical simulations, success in reproducing three-dimensional deformation fields around a non-driving wheelstrongly depended on the value of the friction angle andcohesion. For cohesive soil (u = w = 0), the solutions werenumerically stable. Simulations of rolling on a purely fric-tional (c = 0) material failed, and this illustrates the limita-tions of the particular code and soil constitutive modelchosen. This failure can be attributed to very large defor-mations induced by a rolling wheel and correspondingnear-zero isotropic stresses over a significant portion ofthe material deforming around the wheel. Adding signifi-cant cohesion moves the vertex in the yield condition andflow potential into the tensile zone away from zero isotro-pic stresses, lends stability to the simulations, and prohibitsmaterial avalanching in regions ahead of and to the rear ofthe wheel where it is prone to occur. Effects of dilatancywere not explicitly presented for rolling but were foundto be virtually the same as in indentation [3], with anincrease in dilation angle having a significant influence onvertical and horizontal forces at a given sinkage.

Overall, the numerical and analytic approaches predictsimilar trends in the force–sinkage curves for a rollingwheel, with vertical force increasing with sinkage atdecreasing rate. Quantitative agreement between thenumerical and analytic curves is also reasonable. The theo-retical predictions match the results of small-scale experi-ments very well, and supplemental data from theliterature [47] support this conclusion. Roughly speaking,sinkage is inversely proportional to the width of the wheelfor sandy soils and inversely proportional to the widthsquared for cohesive soils. Thus, reducing wheel width byhalf leads to a two-fold increase in sinkage with sandsand to a four-fold increase in sinkage with clays. For bothtypes of soil, sinkage is nearly inversely proportional towheel diameter, which implies for fixed wheel force atwo-fold increase in sinkage with either soil when wheeldiameter is reduced by half.

Even though stable numerical simulations required sig-nificant cohesion, it appears that the occurrence of local-ized deformation (shear bands or shocks) observed inmany classical geomechanics problems (retaining walls,slopes, foundations) does not take place in wheel rolling.The lack of localized deformation also was observed inthe small-scale experiments on granular material. Geome-chanics problems are often solved or tested as two-dimen-sional (plane strain) problems, while the simulations andexperiments presented in this paper are three-dimensional.It is known (cf. [45]) that plane strain problems are suscep-tible to deformation localization, particularly for non-asso-ciated flow rules, whose adequacy has been generally-accepted in view of tests on specimens of frictional soils.

In presenting the results, attention was focused on thevertical component of wheel force, as it equals weight onthe wheel and appears to be the most important contribu-tor to rutting caused by non-driving wheels. The horizontal(i.e., longitudinal) component of force was also deter-mined, although this force is of interest in studying vehiclemobility and falls outside the scope of this paper.

The results above indicate the ability of the simplified con-stitutive models to reproduce, approximately at least, therelationship between the force on a rolling wheel and wheelsinkage for different soils and moderate sinkage (s/d < 0.1).Predictions closer to experimental data would require moresophisticated soil models. As the emphasis was on the influ-ence of plastic parameters, a limited set of representativeelastic parameters was used in the simulations. The numeri-cal simulations also showed (and the analytic solutionsassume) the existence of a rut and berms left by a rollingwheel. As the volume of berms compensating volumetricallyfor material missing in the rut is small, it may be difficult toassess as whether modeling rut formation necessarilyrequires a constitutive model allowing for compaction ornot. Indeed, it was noted in tests on saturated (nearly incom-pressible) clay that the material outside the rut was elevatedby a hardly discernable amount and spread over a relativelylarge area. This was also observed in the numerical simula-tions of these experiments. Certainly, in very loose soils orsnow, disregarding compaction is unacceptable.

Several conclusions obtained from the analysis of a non-driving wheel closely match those obtained from modelingwheel indentation [3]. Force–sinkage curves have similarshape, and the deformation field is unquestionably three-dimensional in both. Lack of localized deformation wasalso observed in both processes. Of particular interest indetecting weak soils is the observation that the verticalforce in rolling is a predictable fraction, dependent onmaterial properties, of that in indentation. This may pro-vide a practical, first order link between continuous rollingprocesses and the local penetration process.

Acknowledgements

Financial support provided by the Minnesota LocalRoad Research Board and the Shimizu Corporation are


gratefully acknowledged. Computer resources were pro-vided by the Minnesota Supercomputing Institute.

Appendix A

Bearing capacity factors [44,48,49]:

Nq ¼ tan2 p4þ u

2

� �ep tan u;N c ¼ ðNq � 1Þ cot u;N c

¼ 2ðNq þ 1Þ tan u

Shape factors [50]:

F cs ¼ 1þ BL

Nq

Nc; F qs ¼ 1þ B

Ltan u; F cs ¼ 1� 0:4

BL

Depth factors (D/B 6 1) [40]:

F cd ¼ 1þ 0:4DB; F qd ¼ 1þ 2 tan uð1� sin uÞ2 D

B; F cd ¼ 1

Inclination factors [34]:

F ci ¼ F qi ¼ 1� 2bp

� �2

; F ci ¼ 1� bu

� �2

Modified bearing capacity factors [39,40]:

N �c ¼ cot u e�2b tan uN q � 1�

;N �q ¼ N qð1� tan bÞ2;N �c ¼ N cð1� tan bÞ2

Analytic approach; inclined force method:

QV ¼bffiffiffiffiffiffiffiffiffiffiffiffiffids�s2p

�cos

ffiffiffisd

r� �cN c 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

bNq

Nc

!1þ0:07

sffiffiffiffiffiffiffiffiffiffiffiffiffids�s2p

�(

� 1�0:64

ffiffiffisd

r� �2

þ0:17csN q 1þffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

btanu

!

� 1þ0:33tanu 1� sinuð Þ2 sffiffiffiffiffiffiffiffiffiffiffiffiffids�s2p

�1�0:64

ffiffiffisd

r� �2

þcffiffiffiffiffiffiffiffiffiffiffiffiffids�s2p

N c 0:5�0:2

ffiffiffiffiffiffiffiffiffiffiffiffiffids�s2p

b

!1� 1

u

ffiffiffisd

r� �2)

forffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

< b

QV ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

� cos

ffiffiffisd

r� ��cN c 1þ bffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ds� s2p N q

N c

� �1þ 0:07

sb

h i

� 1� 0:64

ffiffiffisd

r� �2

þ 0:17csN q 1þ bffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p tan u

� �

� 1þ 0:33 tan u 1� sin uð Þ2 sb

h i1� 0:64

ffiffiffisd

r� �2

þcbN c 0:5� 0:2bffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ds� s2p

� �1� 1

u

ffiffiffisd

r� �2)

forffiffiffiffiffiffiffiffiffiffiffiffiffiffids� s2p

P bAnalytic approach; inclined footing method:

QV ¼ bffiffiffiffiffidsp

� cos

ffiffiffisd

r� �cN c 1þ

ffiffiffiffiffidsp

bNq

Nc

!1þ 0:07

sffiffiffiffiffidsp cos

�(

�ffiffiffisd

r� ��N qe�2

ffiffisd

ptan u � 1

N q � 1

!

þ 0:17csN q cos

ffiffiffisd

r� �1þ

ffiffiffiffiffidsp

btan u

!"

� 1þ 0:33 tan u 1� sin uð Þ2 cos

ffiffiffisd

r� �sffiffiffiffiffidsp

� �

þcffiffiffiffiffidsp

N c 0:5� 0:2

ffiffiffiffiffidsp

b

!#1� tan

ffiffiffisd

r� � �2)

forffiffiffiffiffidsp

< b and

QV ¼ bffiffiffiffiffidsp� cos

ffiffiffisd

r� �cN c 1þ bffiffiffiffiffi

dsp Nq

Nc

� ��

� 1þ 0:07sb

cos

ffiffiffisd

r� �� Nqe�2

ffiffisd

ptan u � 1

N q � 1

!

þ 0:17csN q cos

ffiffiffisd

r� �1þ bffiffiffiffiffi

dsp tan u

� �

� 1þ 0:33 tan u 1� sin uð Þ2 cos

ffiffiffisd

r� �sb

� �

þcbN c 0:5� 0:2bffiffiffiffiffidsp

� ��1� tan

ffiffiffisd

r� � �2)

forffiffiffiffiffidsp

P b.

References

[1] Li Q, Ayers PD, Anderson AB. Prediction of impacts of wheeledvehicles on terrain. J Terramech 2007;44(2):205–15.

[2] Hambleton JP. Modeling test rolling in clay. MS thesis. Minneapolis:University of Minnesota; 2006.

[3] Hambleton JP, Drescher A. Modeling wheel-induced rutting in soils:indentation. J Terramech 2008;45(6):201–11.

[4] Hambleton JP, Drescher A. Modeling test rolling on cohesivesubgrades. In: Proc of the int conf on advanced characterisation ofpavement and soil engineering materials, vol. 1. Athens, Greece: June2007. p. 359–68.

[5] Bekker MG. Introduction to terrain-vehicle systems. Ann Arbor:University of Michigan Press; 1969.

[6] Karafiath LL, Nowatzki EA. Soil mechanics for off-road vehicleengineering. Clausthal: Trans Tech Publications; 1978.

[7] Wong JY. Theory of ground vehicles. New York: Wiley; 2001.[8] McRae JL. Theory for a towed wheel in soil. J Terramech

1964;1(4):31–53.[9] Maclaurin EB. Soil–vehicle interaction. J Terramech

1987;24(4):281–94.[10] Gee-Clough D. Soil–vehicle interaction (B). J Terramech

1991;28(4):289–96.[11] Muro T. Braking performances of a towed rigid wheel on a soft

ground based on the analysis of soil-compaction. Soils Found1993;33(2):91–104.

[12] Shibly H, Iagnemma K, Dubowsky S. An equivalent soil mechanicsformulation for rigid wheels in deformable terrain, with application toplanetary exploration rovers. J Terramech 2005;42(1):1–13.


[13] Maciejewski J, Jarzebowski A. Experimental analysis of soil defor-mation below a rolling rigid cylinder. J Terramech 2004;41(4):223–41.

[14] Chiroux RC, Foster WA, Johnson CE, Shoop SA, Raper RL. Three-dimensional finite element analysis of soil interaction with a rigidwheel. Appl Math Comp 2005;162(2):707–22.

[15] Shoop SA, Haehnel R, Kestler K, Stebbings K, Alger R. Finite elementanalysis of a wheel rolling in snow. In: Proc of the 10th int conf on coldregions engineering. Lincoln, NH, USA: August 1999. p. 519–30.

[16] Abo-Elnor M, Hamilton R, Boyle JT. 3D Dynamic analysis of soil–tool interaction using the finite element method. J Terramech2003;40(1):51–62.

[17] Karmakar S, Kushwaha RL. Dynamic modeling of soil–tool inter-action: an overview from a fluid flow perspective. J Terramech2006;43(4):411–25.

[18] Plouffe C, Lague C, Tessier S, Richard MJ, McLaughlin NB.Moldboard plow performance in a clay soil: simulations andexperiment. Trans ASAE 1999;42(6):1531–9.

[19] Yong RN, Fattah EA, Boonsinsuk P. Analysis and prediction oftyre–soil interaction and performance using finite elements. J Terra-mech 1978;15(1):43–63.

[20] Block WA, Johnson CE, Bailey AC, Burt EC, Raper RL. Energyanalysis of finite element soil stress prediction. Trans ASAE1994;37(6):1757–62.

[21] Foster WA, Johnson CE, Raper RL, Shoop SA. Soil deformation andstress analysis under a rolling wheel. In: Proc of the 5th NorthAmerican ISTVS conference/workshop. Saskatoon, SK, Canada:May 1995. p. 194–203.

[22] Liu CH, Wong JY. Numerical simulations of tire–soil interactionbased on critical state soil mechanics. J Terramech 1996;33(5):209–21.

[23] Fervers CW. Improved FEM simulation model for tire–soil interac-tion. J Terramech 2004;41(2–3):87–100.

[24] Adebiyi OA, Koike M, Konaka T, Yuzawa S, Kuroishi I. Compac-tion characteristics for the towed and driven conditions of a wheeloperating in an agricultural soil. J Terramech 1991;28(4):371–82.

[25] Chi L, Tessier S, Lague C. Finite element prediction of soilcompaction induced by various running gears. Trans ASAE1993;36(3):629–36.

[26] Arvidsson J, Ristic S. Soil stress and compaction effects for fourtractor tyres. J Terramech 1996;33(5):223–32.

[27] Muro T, He T, Miyoshi M. Effects of a roller and a tracked vehicle onthe compaction of a high lifted decomposed granite sandy soil. JTerramech 1998;35(4):265–93.

[28] Botta GF, Jorajuria D, Draghi LM. Influence of axle load, tyre sizeand configuration on the compaction of a freshly tilled clay. JTerramech 2002;39(1):47–54.

[29] ABAQUS Version 6.6 Documentation. Providence: ABAQUS, Inc.;2004.

[30] Sitkei G. The bulldozing resistance of towed wheels in loose sand. JTerramech 1966;3(2):25–37.

[31] Gee-Clough D. The effect of wheel width on the rolling resistance ofrigid wheels in sand. J Terramech 1979;15(4):161–84.

[32] Godwin RJ, O’Dogherty MJ. Integrated soil tillage force predictionmodels. J Terramech 2007;44(1):3–14.

[33] Terzaghi K. Theoretical soil mechanics. New York: Wiley; 1943.[34] Meyerhof GG. Some recent research on bearing capacity of founda-

tions. Can Geotech J 1963;1(1):16–26.[35] Uffelmann FL. The performance of rigid cylindrical wheels on clay

soil. In: Proc of the 1st int conf on the mechanics of soil–vehiclesystems. Torino, Italy: April 1961.

[36] Onafeko O, Reece AR. Soil stresses and deformations beneath rigidwheels. J Terramech 1967;4(1):59–80.

[37] Krick G. Radial and shear stress distribution under rigid wheels andpneumatic tires operating on yielding soils with consideration of tiredeformation. J Terramech 1969;6(3):73–98.

[38] McRae JL. Theory for a towed wheel in soil. J Terremech1964;1(4):31–53.

[39] Chen WF. Limit analysis and soil plasticity. Amsterdam: Elsevier;1975.

[40] Hansen JB. A revised and extended formula for bearing capacity.Bulletin 28. Copenhagen: Danish Geotechnical Institute; 1970.

[41] Zhu M, Michalowski RL. Shape factors for limit loads on square andrectangular footings. J Geotech Geoenv Eng 2005;131(2):223–31.

[42] Waters AJ, Drescher A. Modeling plug flow in bins/hoppers. PowderTechnol 2000;113(1):168–75.

[43] White DJ, Take WA, Bolton MD. Soil deformation measurementusing particle image velocimetry (PIV) and photogrammetry. Geo-technique 2003;53(7):619–731.

[44] Prandtl L. Uber die Eindringungsfestigkeit (Harte) plastischerBaustoffe und die Festigkeit von Schneiden. Zeit angew Math Mech1921;1(1):15–20.

[45] Vardoulakis I, Sulem J. Bifurcation analysis in geomechanics.London: Chapman and Hall; 1995.

[46] Drescher A, Michalowski RL. Density variation in pseudo-steadyplastic flow of granular media. Geotechnique 1984;34(1):1–10.

[47] Willis BMD, Barret FM, Shaw GJ. An investigation into rollingresistance theories for towed rigid wheels. J Terramech1965;2(1):24–53.

[48] Reissner H. Zum Erddruckproblem. In: Proc of the 1st int congressfor applied mechanics. Delft, The Netherlands: 1924. p. 295–311.

[49] Vesic AS. Analysis of ultimate loads of shallow foundations. J SoilMech Found Div ASCE 1970;99(1):45–73.

[50] De Beer EE. Experimental determination of the shape factorsand bearing capacity factors of sand. Geotechnique1970;20(4):387–411.

modeling wheel-induced rutting in soils: rolling · 2013. 1. 22. · theoretical models for...

Documents